Renewables and Storage in Distribution Systems: Centralized vs. Decentralized Integration

0.2 MB·

by  Liyan Jia

1
Renewables and Storage in Distribution Systems:
Centralized vs. Decentralized Integration
Liyan Jia and Lang Tong, Fellow, IEEE
6 Abstract—The problem of integrating renewables and storage home energy storage. The possibility that energy storage is
1 into a distribution network is considered under two integration omnipresent in the power grid has profound implications:
0 models:(i)acentralizedmodelinvolvingaretailutilitythatowns demand no longer needs to be balanced instantaneously and
2 theintegrationaspartofitsportfolioofenergyresources,and(ii) volatilities due to renewables drastically smoothed out. A
adecentralizedmodelinwhicheachconsumerindividuallyowns
n and operates the integration and is capable of selling surplus relevantpointof referenceis the roleof bufferin the Internet.
a electricity back to the retailer in a net-metering setting. It is precisely the omnipresence of such memory elements
J
The two integration models are analyzed using a Stackelberg throughout the Internet that makes it possible to stream high

1 game in which the utility is the leader in setting the retail
definition video in real-time over unreliable and stochastic
1 price of electricity, and each consumer schedules its demand
networking environments.
by maximizing individual consumer surplus. The solution of the
] Stackelberg game defines the Pareto front that characterizes The other premise called into question is the notion that
C
fundamental tradeoffs between retail profit of the utility and the transmission grid is the central artery that delivers power
O consumer surplus. fromgenerationsitestomillionsofendusers.Thearchitecture
It is shown that, for both integration models, the centralized
. of power delivery is a unidirectional transfer of power from
h integration uniformly improves retail profit. As the level of
generators through transmission networks to distribution net-
t integration increases, the proportion of benefits goes to the con-
a sumers increases. In contrast, the consumer-based decentralized worksandtoconsumers.Suchanoperationparadigmrequires
m
integration improves consumer surplus at the expense of retail overprovisioningof generationand transmissioncapacitiesto
[ profit of the utility. For a profit regulated utility, the consumer- ensurethatextremedemandsaremetundercontingencies.The

1 based integration may lead to smaller consumer surplus than cost of over-provision, however, is enormous. The difference
that when no renewable or storage is integrated at either the
v betweenthe highandthe lowhourlypricesoverthecourseof
consumer or the retailer end.
5 a typical year can be up to four orders of magnitude [1].
1 IndexTerms—Distributedenergyresources,microgrid,renew-
The premise of unidirectional power delivery is also chal-
3 able integration home energy storage, dynamic pricing, game
2 theory in demand response. lengedby thephenomenalgrowthinphotovoltaic(PV)instal-
0 lationsat the consumerend, drivenby the precipitousdrop of
1. I. INTRODUCTION PVcostandpoliciesthatpromotescleanenergy.Akeypolicy
isthewidelyadoptednet-meteringmechanismthateffectively
0 The power grid is facing an imminent transformation,
6 allows the consumer to sell back locally generated surplus
perhaps one of the most profound in its 130 years history.
1 energy at the retail price. As a result, power generated at the
This transformation is brought by disruptive innovations that
: consumer end becomes a potential source of overall power
v challengebasicpremisesthatgovernsitsdesignandoperation.
i supply, resulting in a new paradigm of power delivery based
X One of the premises is the long-held axiom that the op-
onabidirectionalmodelwhereasubstantialpartofthepower
eration of the grid is governed by the law of instantaneous
r generation may come from the consumer end.
a balance of generation and consumption.Thus the principle of
The transformative innovations of large scale PV and stor-
operationistoforecastdemandaccurately,secureinexpensive
age present new engineering and social economic challenges.
generation resources, and procure sufficient reserves to cope
At the core of these challenges is the complex question on
with uncertainties. The resulting approaches, unfortunately,
howlargescale renewablesandstorageare integratedintothe
often result in complex, ad hoc, and costly practices. Indeed,
overall power delivery. Answers to such questions have sig-
the requirement of instantaneous power balance is one of the
nificant impacts on architectures of control, communications,
fundamentalbarriersto integratinga highlevelof intermittent
and markets for the future power grid.
renewables that defy accurate forecast.
The axiom of instantaneous power balance is challenged
by the recent introduction of commercial grade batteries for A. The Death Spiral Hypothesis and Integration Models
Liyan Jia and Lang Tong are with the School of Electrical The rise of renewable integration by consumers has led to
and Computer Engineering, Cornell University, Ithaca, NY, USA theso-calleddeathspiralhypothesisforretailutilities[2],[3]:
{lj92,lt35}@cornell.edu
the decline of consumptiondue to below-the-meterrenewable
PartofthisworkwaspresentedatIEEEPESGeneralMeetings,July,2015.
This work is supported inpart by the National Science Foundation under generation by consumers triggers an upward pressure on the
GrantCNS-1135844andGrant1549989. rate of electricity and threatens the economic viability of
This manuscript has been accepted by IEEE Journal on Selected Areas
the retail utility. As a regulated monopoly, the retail utility
in Communications - 2016 Special Issue on Emerging Technologies in
Communications. ultimately has to raise the rate of electricity, which acceler-
2
ates consumer adoptions of distributed generation that further participationin a smartgrid,pricingof electricity thatreflects
erodes the revenue of the utility. the overallsystem demandandsupplyplaysa centralrole.To
Alternative rate structures have already been proposed that this end, the notion of real-time pricing (RTP) has received
makes connection charges a more significant part of the cost considerable attention since the deregulation of electricity
for the consumers. Such increases erect a seemingly artificial market. RTP allows the price of electricity to vary with time,
barrier that inhibits renewable integration. If implemented in- and it has been widely argued that RTP is critical for an
discriminately,itshiftsthecosttothosewithoutthecapability efficient electricity market [4]. It was pointed out in [5] that
of below-the-meter renewable generation. RTP brings significant gain in efficiency even if “demand
Insights into the death spiral hypothesis can be gained by shows very little elasticity.” When renewable generation is
a better understanding of integration models in distribution considered,studiesshowthatRTPcanincreasethepercentage
systems. To this end, we consider two types of integrations; of load served by renewables even when the demand has low
oneisthebelow-the-meterintegrationbyconsumers,theother elasticity [6].
is the above-the-meter integration by the utility∗. Inthispaper,weconsidera specialformofRTPreferredto
For the utility-based integration, the utility owns the in- as day ahead hourly pricing (DAHP). DAHP sets the price
tegration resources†. The operation decisions are centralized of electricity for retail consumers one day ahead, possibly
with a global objective, unobservable to the consumers. The in conjunction with the clearing process of the day ahead
integrator can coordinate resources within its service area, wholesale market. The rationale is that DAHP provides a
mitigating uncertainties inherentin renewable integration. We level of price certainty that allows a consumer to plan her
thus refer this model to as centralized integration. consumption accordingly. It also avoids potential volatility
Fortheconsumer-basedintegration,consumersowntheen- and instability when RTP is determined based on the real-
ergyresourcesandoperatetheintegrationandstoragedevices. time locational marginal prices (LMP) [7]. DAHP has been
The integrationis thereforedecentralizeddrivenbyindividual implemented by several utility companies in the U.S. [8].
objectives of consumers. In this case, renewable integration Itisnaturaltomodelconsumer-retailerinteractioninagame
and storage operations are below the meter, unobservable to theoretic setting; the retailer sets the price of electricity and
the retail utility. We call this modeldecentralized integration. the consumer adjusts the consumption accordingly. This is
Itisintuitivethatdecentralizedintegrationmayhaveadverse the Stackelberg game model adopted in this paper and it has
effectontherevenueoftheutility.Doesitbenefitconsumers? been used earlier. In particular, the authors of [9] used the
The answer may be less obvious and more nuanced. To a Stackelberg game to develop an adaptive pricing scheme that
consumer who still relies on the grid to deliver power at maximizes social welfare. The work of Chen et al. [10] and
times ofshortage,it is notclear whetherthe benefitof locally Li et al. [11] consideredthe problem of setting retail price of
generated renewables offsets the cost increase of electricity electricityviadistributedsocialwelfareoptimizationandequi-
purchased from the retailer. For the centralized integration, it librium analysis. These results characterize consumer-retailer
is not clear whether the utility will share the derived benefit interaction under the setting of social welfare maximization.
with consumers such that consumers find it economically They are not adequate for analyzing the consumer-retailer
attractive to stay with the utility’s integration model rather interaction when their interests are not well aligned.
than switching to the consumer-based integration; the latter 2) Consumer-retailer interaction with renewable integra-
creates a drift toward the death spiral. tion: Renewable integration in distribution systems has pro-
foundimplicationsonconsumer-retailerinteractions;thedeath
spiral hypothesis discussed earlier is one such example. The
B. Related work
first study of the death spiral hypothesis is by Cai et al. [2].
The literature on renewable integration and storage in dis-
The empirical study is based on a model that captures the
tribution systems is broad and extensive. Here we highlight
closedloopdynamicsofconsumeradoption(attheyearlytime
some of the relevant literature that focuses on interactions
scale). The cost of net-metering and the effect of connection
between consumers and a distribution utility in the presence
charge are also considered.
of renewable generation and storage in distribution systems.
A relevant line of work is pricing of electricity in the
Hereafter, we use the terms distribution utility and retailer
presence of renewables. The authors of [12] considered the
interchangeably as we consider the retail functionality of the
problemof contractdesign betweenrenewablegeneratorsand
distribution utility.
energyaggregatorswhoareresponsibleforalargescalePHEV
Our discussion centers around three broad areas: (i) retail
charging. The work in [13] shows that, by coupling with
pricing of electricity; (ii) consumer-retailer interaction with
deferrableloads,thecostsresultingfromstochasticgeneration
renewable integration; (iii) consumer-retailer interaction with
can be significantly mitigated. In [14], [15], a stochastic
storage integration.
optimization framework for retail pricing is formulated that
1) Retail Pricing and Real-time Price (RTP): The inter-
takes into account uncertainty associated with the wholesale
action between the retailer and consumers is fundamentally
marketandtheuncertaintyassociatedwithconsumerdemand.
defined through the pricing of consumption. For consumer
3) Consumer-retailerinteractionwith storage integration:
There has been broad interest in the role of storage in
∗Here we use utility to include distribution utility as well as energy
distribution networks, especially when it is coupled with the
aggregators.
†Theutility mayalsocontract renewable providers andstorageoperators. charging of electric vehicles. The existing literature focuses
3
mostly on the optimal management of storage usage and the B. Consumer Surplus and Optimal Demand Response
economic benefits of storage. A particularly relevant work is
The benefit to consumer is measured by the consumer
[16]wheretheauthorspointedoutthatthepresenceofstorage
surplus (CS) defined by the utility of consumption minus the
may result in gains in consumer surplus, losses in generator
cost. We are interested in characterizing the demand function
surpluses, and net welfare gain. Incentives for storage usage
associated the optimal demand response to DAHP. To this
by the merchantstorage operators, generators,and consumers
end, we consider a linear demand model (Theorem 1) that
and structures of storage ownership are analyzed in [17].
arises from the thermostatically controlled load (TCL) where
There has been a considerable amount of work on optimal
the energy state (temperature) is controlled by thermostatic
storage management under various settings. See [18] and
controlled heating-ventilation and air conditioning (HVAC)
references therein. The problem is in some way similar to
units.
the inventory control problem but has special characteristics
Empirical study [24] shows that temperature evolution can
unique to the distribution and consumption of electricity. In
be modeled by a discrete-time dynamic equation given by
particular,athresholdstructureoftheoptimalpolicyofstorage
operationis obtained and analyzedin [18]. Similar results are xt =xt−1+α(at−xt−1)−βpt+wt, (1)
also obtained in [19] under more general pricing and demand yt =(xt,at)+vt,
models.Theauthorsof[20]consideredtheinterestingproblem where x is the indoor energy state (temperature), a the
t t
of cooperative or competitive operations between storage and outdoor temperature, p the power drawn by the HVAC unit,
t
renewablegenerations.Theseexistingresults,however,donot y the temperature measurement, w and v the process and
t t t
address directly the impact of storage on consumer-retailer measurement noise, respectively, that are assumed to be zero
interactions, which is the main focus of the current paper. mean and Gaussian.
System parameters α ∈ (0,1) and β model the insolation
C. Summary of results and notations
of the building and efficiency of the HVAC unit. The above
This paper focuses on how economic benefits of renewable equation applies to both heating and cooling scenarios. But
and storage integration are distributed between the utility and we excludethe scenariothattheHVACdoesbothheatingand
the consumers, and how different integration models affect cooling during the same day, focusing herein on the cooling
such distributions. The analytical framework presented here scenario (β >0) without loss of generality.
was first proposed in [21] and further developed in [22]. The Assuming that a consumer’s discomfort level can be mea-
applicationofthisframeworktorenewableintegrationwasfirst sured by the squared deviation of energy state from her
presented in a conference version of the paper [23] and with preferred setting, the CS for consumer k over a period of N
additional material added here. The analysis and numerical intervals is given by
resultsofstorageintegrationundertwo integrationmodelsare
N
new. cs(k)(π)=−µ(k) (x(k)−θ(k))2−πTp(k), (2)
Standard notations are used throughout the paper. We use t t
x+ to representmax{0,x},x− to representmax{0,−x}.For Xt=1
avectorx, xT isitstranspose.Herex=(x ,··· ,x )isfora where π is the DAHP vector, p(k) the vector of power usage
1 N
column vector and [x ,··· ,x ] a row vector. For a random over N periods, x(k) and θ(k) are the the actual and desired
1 M t t
variable x, x¯=∆E(x) is the expected value. energystates, respectively,in intervalt, and µ(k) a coefficient
that converts the level of discomfortto some monetary value.
II. SYSTEMMODEL Note that π should contain prices at the same time scale
We assume that the retail utility participates in a two- as the system dynamics (by duplicating hourly prices). To
settlementwholesalemarketasaloadservingentity;itsubmits simplify notation, we assume that the time t is at the hourly
bids in the day ahead market that sets the day ahead price, scale.Theresultsherecanbegeneralizedeasily.See[22].Note
and it is a price taker in the real-time market. The retailer also that CS defined here is always negative.It represents the
determines the day ahead hourly price (DAHP) and makes it totalequivalentcost( actualcostplusthe costconvertedfrom
availableto the consumer.The consumeradjustsits consump- discomfortlevel)eachconsumerneedstopayoverN periods.
tion in real time. GivenDAHPπ,theoptimaldemandresponseforconsumer
k is to maximize its expected CS with respect to the random
A. Dynamic Pricing noises w and v,
DAHP is defined a 24 dimensional vector π posted one
max E −µ(k) N (x(k)−θ(k))2−πTp(k)
day ahead and fixed throughout the day of operation; it is a p(k) t=1 i t
generalization of some of the existing pricing schemes: the s.t. x(tnk) =x(t−kP)1+α(k)(at−x(t−k)1)−β(k)po(tk)+wt(k),
uniformpricing is when entriesof π are the same;the critical y(k) =(x(k),a )+v(k),
t t t t
peakpricing(CPP)andthetimeofuse(TOU)pricingiswhen (3)
entriesofπ correspondingtothepeakhoursaredifferentfrom where the expectation is taken over the observation and
those in off-peak hours. The retailer can also index DAHP to process noises. Here we ignore the positivity constraint on
the day ahead prices at the wholesale market. In this paper, p(k),andassumethattheresultingoptimalpowerconsumption
we are interested in optimized DAHP by the retailer under a is positive. This assumption is reasonable when the DAHP
general notion of social welfare. doesn’t vary too much during a day and µ is large. The
4
following theorem gives the aggregated demand function and
RetailProfit(RP)
the consumer surplus summed over all consumers.
Theorem 1 (Optimal Demand Response [22]): Assume πo
that the process noise is Gaussian with zero mean for all πr
consumers.GivenDAHPπ,theoptimalaggregatedresidential ∆
real-time demand and the expected total CS are given by,
respectively,
πsw
dRT(π) = p(k) =b−Gπ, (4)
ConsumerSurplus(CS)
k
X
cs(π) = πTGπ/2−πT¯b+c, (5)
Fig. 1: CS-RP trade-offs without integration and the Pareto
front. The point associated πsw is the social welfare
where matrix G ≥ 0 is deterministic and positive semi- optimal tradeoff, πo the profit maximizing tradeoff,
definite, and it depends only on system parameters and user and πr the profit regulated tradeoff associated with
preferences. Parameter b is a Gaussian vector independent of retail profit ∆.
π, and c is a deterministic constant, also independent of π.
Hereafter, our analysis is based on the optimal demand
response of the form in (4) and independentof how (4) arises
Byvaryingη,weobtaintheParetofrontofCS-RPtradeoffs
from the types of demand response applications.
defined by
C. Retail Profit P= cs(π∗(η)),rp(π∗(η)) :η ∈[0,1] , (9)
( )
Theretailprofit(RP)fortheutilityisthedifferencebetween (cid:16) (cid:17)
the revenue from its customers and the cost of power. Let whereeachpointontheParetofrontrepresentsanoptimalCS
λ = (λ ,λ ,...,λ ) be the random vector of wholesale spot vs. RP tradeoff associated with an optimal DAHP π∗(η) as a
1 2 T
pricesofelectricityattimesofconsumption,whichrepresents solutionof(8).Asafunctionofπ,theweightedsocialwelfare
is given by
the marginal cost of procuring electricity from the wholesale
market.Inabsenceofrenewablesaccessibletotheretailer,the sw (π)=(η −1)πTGπ+πT((1−η)¯b+Gλ¯)+ηc, (10)
retail profit rp(π) is given by η 2
from which we obtain the optimal DAHP as
rp(π)=(π−λ)TdRT, (6)
1 1−η
π∗(η)= λ¯+ G−1¯b, (11)
where dRT and λ are random and π the decision variable 2−η 2−η
representing the utility’s action.
where the first term depends only on the expected wholesale
As a load serving entity, the utility takes into consideration
priceofelectricityλ¯. Thistermshowsthatthe optimalDAHP
its ownprofitandconsumers’satisfaction. Here we define the
followstheaveragewholesaleprice.Thesecondtermdepends
weighted expected social welfare as
only on the consumer preference and randomness associated
sw (π)=rp(π)+ηcs(π), (7) with consumer environments.
η
Fig 1 illustrates various CS-RP tradeoffs. The shaded area
where the expectation is taken over randomness in the real represents the achievable region of CS-RP pairs by suitably
time wholesale price of electricity λ and the randomness in choices of DAHP, and the boundary of the shaded regions is
the consumer demand in (4). When η = 0, (7) corresponds the Pareto front of all valid tradeoffs.
to the profit maximization and η = 1 the social-welfare Various properties of the achievable region of the CS-
maximization. RP tradeoff have been established in [22]. In particular, the
achievable region is convex, and Pareto front concave and
decreasing. A particularly important point on the Pareto front
III. A STACKELBERG GAME MODELAND ITS SOLUTION
corresponds to the social welfare optimizing pricing πsw. It
Itisnaturaltomodelutility-consumerinteractionviaDAHP has been shown that πsw = λ¯, and the resulting retail profit
by a Stackelberg game where the utility is the leader and the iszero[22],whichimpliesthatthesocial-welfaremaximizing
consumers are the followers; the action of the leader is to objectiveisnoteconomicallyviablefortheutility.Iftheutility
set DAHP π one day ahead, and followers respond with real has to operate at some level of retail profit ∆ as a regulated
timeconsumption.Thepayoffsfortheconsumeriscs(π),and monopoly, it is to the utility’s interest to maximize consumer
payoff to the utility is swη(π). surplus by setting DAHP at πr.
For a fixed η, the Stackelberg equilibrium is obtained
via backward induction in which the optimal response of a
IV. RENEWABLEINTEGRATION
follower is given in Theorem 1, and the optimal action of the
In this section, we discuss how renewable integration
leader is to set DAHP optimally by
changes the characteristics of the Pareto front of the CS-RP
π∗(η)=argmax {rp(π)+ηcs(π)}. (8) tradeoff under the centralized and decentralized integration
5
Retail with renewable Consider first the retail profit maximizing point on the
Retail
profit without renewable profit without renewable Pareto front. Because the renewable integration is below the
meter, the retailer simply experiences the reduced consump-
with renewable
tion, and the maximum retail profit decreases. For the social
welfare maximizing point, because the retail profit is zero,
gain Loss the renewable integration benefits entirely to the consumer.
consumer consumer We thus expect that the new social welfare optimizing price
(a) surplus (b) surplus
results in increased consumer surplus. Interpolating the two
points,the correspondingParetofrontmustintersectwith that
Fig. 2: CS-RP tradeoffs with renewable integration: (a)
when there is no renewable.
centralized integration; (b) decentralized integration.
Thecaseofprofitregulatedutilityisparticularlyinteresting.
Dependingonthelevelofregulatedretailprofit,decentralized
models. We first provide intuitions of these results followed renewable integration by the consumer may lead to reduced
by detailed analysis. consumer surplus as illustrated in Fig. 2(b). The reason is
For the simulations in this section, we used the actual that, the renewable integration below the meter reduces the
temperature record in Hartford, CT, from July 1st, 2012 to consumption, which forces the utility to increase the retail
July 30th, 2012. The day-head price (used as prediction) and price to cover the loss of revenue. If the energy cost increase
real-time price (used as realization) were also for the same isgreaterthantheutilityofconsumption,theconsumersurplus
periodfromISONewEngland.TheHVACparametersforthe is reduced.
simulationwassetas:α=0.5,β =0.1,µ=0.5.Thedesired
indoortemperaturewas set to be 18◦C for all hours.The size B. Analysis of Centralized Renewable Integration
of total consumers is 1000.
We now analyze the Pareto frontwhen the utility ownsand
operates centrally renewable integration. We assume that the
A. General characteristics and intuitions
amountofrenewableisnotlargeenoughtosatisfyalldemand
A graphical sketch of the main results is shown in Fig. 2 of its service area, or equivalently,any excess of renewable is
where the left panel compares the Pareto fronts with and spilled.
without renewables under the centralized integration, and the For simplicity, we assume zero cost of using renewable
right panel is the comparison under decentralized integration. energy and denote the renewable available for the retailer in
a) Centralized integration: For the centralized integra- each hour as a random vector q = (q ,...,q ). The retail
1 24
tion, as expected, the Pareto front is strictly above that of no profit after renewable integration then becomes
renewable integration, as shown in Fig. 2(a). Note that if the
rp(π)=πTdRT−λT(dRT−q)+. (12)
utilityoperatesatafixedretailprofitasaregulatedmonopoly,
renewable integration by the utility results in increased con- The CS-RP region achievable by DAHP will be enlarged by
sumer surplus. renewable integration, obviously. What is less obvious is to
Our result further quantifies the distribution of benefits whatdegree renewablebenefitsconsumers.If the Pareto front
among the utility and the consumers. In particular, we show inFig2(a)movesstraightupwards,thentheutilitycapturesall
in Theorem 2 that, the benefit of renewables goes to the con- the benefit. If the Pareto front shifts up right, both the utility
sumers only if the available renewable capacity K exceeds a and the consumer benefit at perhaps different degrees.
certainthreshold.AsK increases,thebenefitsfromrenewable The following theorem characterizes how the benefits are
integration apportioned to consumer increases monotonically. distributed between the utility and the consumers for deter-
The intuition behind the threshold behavior is as follows. ministic demand function.
When the amount of renewables is small, the utility simply Theorem 2 (Centralized Renewable Integration): Assume
uses the available renewables to reduce the amount of pur- that q are independently distributed over [0,KΓ ] and
i i
chased power from the grid. In this regime, the Pareto front the demand function associated with the optimal demand
shifts straightlyupward,which meansthat all benefitsgoesto response is deterministic (b = ¯b). Let ∆rp(η) and ∆cs(η)
the utility. As the amount of renewable increases, it becomes be the increase of retail profit and consumer surplus due to
necessary that the utility reduces the retail price to stimulate renewable integration for fixed η, respectively. Then,
consumption.Therefore,beyondthethreshold,theParetofront
1) for all η, ∆rp(η)>0.
shifts upright; the utility and the consumer split the profit.
2) Thereexistsaτ suchthattheconsumersurplusimproves
only when K >τ. Specifically, ∆cs(η)=0 for K ≤τ
and ∆cs(η)>0 otherwise.
b) Decentralized integration: For the consumer-based
3) The fraction of renewable integration benefit to the
integration, we show in Theorem 3 that the Pareto front with consumers ∆cs(η) → 1 as K →∞.
renewable integration intersects with that when there is no ∆cs(η)+∆rp(η) 3−2η
Proof:Withoutrenewableintegration,foraparticularη,the
renewableintegration,as illustrated in Fig. 2(b). The intuition
first order condition of Eq. (8) gives that the optimal price
for this behavior comes from examining the points on the
π∗(η) satisfies
Pareto front corresponding to the social welfare maximizing
price and the retail profit maximizing price. (η−2)Gπ∗(η)+(1−η)¯b+Gλ¯ =0.
6
1
3)0 6 1MW renewable
1 5 )0.8 3MW renewable
( η
ofit 4 η)rp∆(0.6 51M0MWW r erneenwewaablbele
ailpr 23 01 MreWne rwenaebwleable cs∆(sη()+0.4
Ret 1 3MW renewable c∆0.2
0 −10 C−8onsume−r6surplus−(4103) −2 00 0.2 0.4 η 0.6 0.8 1
Fig. 3: CS-RP Tradeoff: centralized integration of Fig. 4: Fraction of renewable benefit to consumers in
renewables centralized integration
Since optimal demand level satisfies d¯(η) = ¯b−Gπ(η), the Fig. 3 shows an numerical example when the capacity of
previous condition is equivalent to the aggregated renewable is small. In this case, the trade-
¯b−(2−η)d¯(η)=Gλ¯. off curve goes directly up; all the benefit from renewable
integration goes to the utility side. On the other hand, when
With renewable integration, the utility optimizes the capacity is large, the trade-off curve goes upright, which
meanstherenewableintegrationbenefitissharedbytheutility
sw (π) = πTd¯−λ¯TE(d¯−q)+)]
η and consumers.
+η(πTGπ/2−πT¯b+c) In Fig. 4, we plotted the distribution of renewable integra-
= ¯bTG−1d¯−d¯TG−1d¯ tion benefits with different renewable integration levels. We
d¯i can see that as the level of integration (K) increases, the
− λ¯i (d¯i−qi)fi(qi)dqi fraction of benefit to the consumers, ∆cs(η), also increases,
Xi Z0 convergingasymptotically to 1 as K∆rpi(nηc)reases.
+η(d¯TG−1d¯/2−¯bTG−1¯b/2+c) (13) 3−2η
where d¯= ¯b−Gπ and f(q) = (f (q ),...,f (q )) is the C. Analysis of Decentralized Renewable Integration
1 1 24 24
marginal PDF of the renewables. We can have the first order
In this section, we analyze the scenario that the consumers
condition with respect to the optimal demand level d¯r(η), haveaccesstorenewableenergy,modeledasa24-dimensional
which satisfies
randomnonnegativevector,s=(s ,s ,...,s )withthemean
1 2 24
¯b−(2−η)d¯r(η)=G(λ¯◦F(d¯r(η))), (14) s¯. The aggregated demand dRsT is therefore
where◦meanstheHadamardproduct,i.e., piecewiseproduct dRsT =b−s−Gπ. (15)
of two vectors. F = (F1,...,F24), where Fi is the marginal Here we assume individual consumers can sell surplus power
cdf of renewable energy’s distribution at hour i. backto the retailer.But we restrictto the case thatthe overall
If for all i, KΓi ≤ d¯i(η), Fi(d¯i(η)) = 1, we can renewable integration level is not high so that the aggregated
see that d¯(η) satisfies the optimal condition (14). Therefore demand dRT ≥ 0 everywhere. Therefore, the utility does not
d¯r(η)=d¯(η), πr(η)=π(η), the change of consumer surplus offer negastive prices.
∆cs(η) = 0. Otherwise, if for some i, KΓi > d¯i(η), Accordingly, the expected consumer surplus is changed to
cFoin(dd¯ii(tiηo)n) (<14)1..DTehneoretefodr¯er(dη¯()η)=ddo¯(eηs)n+otGsaδt/is(f2y−thηe),opwtihmerael c(πss−(πλ¯))=Ts¯.csT(hπe)f+olπloTws¯,inagndthtehoereremtaiclhparroacfitterripzse(sπt)h=e srhpa(pπe)o−f
δ =λ¯−λ¯◦F(d¯r(η))≥0. Then, the new Pareto front.
∆cs(η) = 1{(d¯(η))TG−1d¯(η)−(d¯(η))TG−1d¯(η)}
2 r r
= 1{2δTd¯(η)+δTGδ}>0.
2 Theorem 3 (Decentralized renewable integration): Denote
For the retail profit, for all K and η, the social welfare and retail profit maximization prices as πo
s
∆rp(η) = (πT(η)d¯(η)−λ¯T(d¯(η)−q)+) and πsw, respectively.
r r r s
−(πT(η)d¯(η)−λ¯Td¯(η)) 1) Social welfare maximization price for the decentralized
= (1−η){(d¯r(η))TG−1d¯r(η)−(d¯(η))TG−1d¯(η)} integration is πssw = λ¯ retail profit is rps(λ¯) = 0. The
+21K{(d¯r(η))TΛ¯d¯r(η)}>0, social welfare maximization point (css(πssw),rps(πssw))
is outside the original CS-RP trade-off curve.
where Λ¯ = diag(λ¯ /Γ ,...λ¯ /Γ ). As K goes
1 1 24 24 2) Maximized retail profit rp (πo)) is smaller than maxi-
to infinity, d¯(η) is bounded, ∆rp(η) goes to s s
r mized retail profit without renewable integration.
(1 − η){(d¯(η))TG−1d¯(η) − (d¯(η))TG−1d¯(η)}, ∆cs(η)
r r 3) As s¯ increases, the maximized social welfare increases
equals to 1{(d¯(η))TG−1d¯(η) − (d¯(η))TG−1d¯(η)}, then
2 r r and the maximized retail profit decreases.
∆cs(η) goes to 1 . (cid:4)
∆cs(η)+∆rp(η) 3−2η Proof: When η = 1, sw (π) = sw(π)+λ¯Ts¯. Therefore,
s
the social welfare maximization price is still λ¯ and resulted
7
Retail with storage Retail
3)5 0 renewable profit without storage profit without storage
0 1MW renewable
14 3MW renewable with storage
(
fit3
o
r
p gain Loss
2
ail
Ret1 (a) csounrpsluumser (b) csounrpsluumser
−08 −7 −6 −5 −4 −3 −2 −1
Consumer surplus (103) Fig. 7: CS-RP tradeoffs with storage: (a) centralized
integration; (b) decentralized integration.
Fig. 5: CS-RP trade-off: decentralized renewable
integration.
V. STORAGE INTEGRATION
6
We now consider storage integration under the centralized
) without renewable
305 utility side renewable and decentralized integration models. The problem and the
(14 consumer side renewable technique of analysis are similar to that used in renewable
fit
o 3 integration except that we need to obtain the optimal policy
r
p
of storage operation.
etail 12 For the simulations in this section, we used the same basic
R setup as Section IV. For storage, we set the storage size as
−08 −7 −6 −5 −4 −3 −2 −1 10KW for individual houses, initial storage level as 0, all
Consumer surplus (103)
efficiency coefficient as 0.95 and ramp limit as 5KW.
Fig. 6: CS-RP trade-off curve comparison with
utility-based and consumer-based renewable
A. General characteristics and intuitions
Aswewillshowinthefollowingtwosubsections,thechar-
retail profit is 0. Since λ¯Ts¯ > 0, the social welfare max- acteristicsofcentralizedanddecentralizedmodelsforstorage,
imization point, (0,sws(λ¯)), is outside the original CS-RP as shown in Fig. 7, are similar to renewable integration.
trade-off curve without renewable energy. As s¯increases, the For the centralized model, the CS-RP tradeoff region is
maximized social welfare also increases. enlarged as shown in Fig 7(a). Different from the case in
When η = 0, the payoff function is the retail profit, renewable integration, the Pareto front is shifted strictly up,
rp (π)=(π−λ¯)T(¯b−s¯−Gπ). Therefore, the optimal price indicatingthatthebenefitofstoragegoesentirelytotheutility.
s
is πo = 1(G−1¯b−G−1s¯+λ¯). The retail profit change with This is due to the fact that, under centralized integration, the
s 2
renewable energy is, optimal use of storage by the utility is to arbitrage electricity
in the wholesale market, independent of consumer response.
1
∆rp=rp(πo)−rp (πo)=− (G−1¯b−λ)Ts¯. The gain from storage integration all goes to the utility.
s s 2
Forthedecentralizedintegration,asinFig7(b),undersome
SinceG−1¯bisthepricetomakedemandequaltozero,G−1¯b− conditions (as shown in Theorem 4), the tradeoff curve inter-
λ≥ 0. Therefore, ∆rp <0. The maximized profit decreases. sectswiththeoriginalone,similartothecaseofdecentralized
Also, we can see that the maximized retail profit decreases integrationofrenewables.Thereasonsbehindthisissomewhat
with the increase of renewable energy level s¯. (cid:4) different from those in renewable integration, although the
method of analysis is similar. Since the consumers have the
Fig. 5 shows numerical results under different integration
abilitytoshifttheirenergyusage,theycanreducethepayment
levels of renewable. It is apparent that the Pareto fronts at
by charging during the low price hours and discharging
1MWand3MWhavecrossoverswiththethatassociatedwith
during the high price hours. Therefore the maximized retail
norenewableintegration.Itisalsoevidentthat,asthelevelof
profit is decreased. On the other hand, in the social welfare
renewables increases, the maximized retail profit decreases,
maximization case, the utility passes the distribution cost to
the maximized social welfare increases, and the cross point
the consumers. In this case, the storage is used to arbitrage
moves to the right along the original trade-off curve.
over the cost to increase the social welfare. Therefore, the
new tradeoff curve intersects with the previous one.
We also compared distributed and centralized control of
renewableenergywiththesamelevelofintegration.Asshown
in Fig. 6, the tradeoff curve with utility-based renewable is
B. Analysis of Centralized Storage Integration
completely outside the one with consumer-based integration,
Denote the storage level at hour i as B , and the energy
which means that utility-based renewable brings more benefit i
chargedintothestorageasr (whenr ≤0,itmeansdischarg-
to the retail market, in absence of other considerations. i i
ing the storage). To differentiate charging and discharging
scenarios,weuser+ ≥0andr− ≥0torepresentthepositive
i i
8
and negativepart of r , i.e., r =r+−r−, respectively.The Using the same notation for storage as in Section V-B,
i i i i
dynamics of the battery can be expressed as the optimaldemandresponseproblemforthe consumerswith
decentralized storage is changed to
B =κ(B +τr+−r−/ρ), (16)
i+1 i i i max E {−µ N (x −θ )2−πT(p+r+−r−)
where κ ∈ (0,1) is the storage efficiency, τ ∈ (0,1) the p,r+,r−,B w,v t=1 t t
−ψTr+−ψTr−}
charging efficiency, and ρ∈(0,1) the discharging efficiency. + P−
s.t. x =x +α(a −x )−βp +w ,
Retail profitof the utility is the differencebetween revenue t t−1 t t−1 t t
y =(x ,a )+v ,
andcost. Forrevenue,centralizedstorageintegrationdoesnot t t t t
B =κ(B +τr+−r−/ρ),
affect the optimaldemand response by consumers. Therefore, t+1 t t t
B =B ,0≤B ≤C,
for fixed DAHP π, the revenue for the utility is the same 24 0 t
0≤r+ ≤ru,0≤r− ≤rd.
withorwithoutstorageintegration.Ontheotherhand,storage t t
integration allows the utility arbitrage over the wholesale Under the net-metering assumption (see [19] for more
price to reduce cost, thus increasing the overall retail profit. general results), the above optimization can be divided into
Formally, the retail profit with centralized storage integration, two independent sub problems: optimizing demand response
rp (π), can be expressed as, asifstoragedoesnotexistandoptimizingstorageforarbitrage
cs
as if there is no demand. This means that adding storage on
rp (π)= max πTd¯−λ¯T(d¯+r+−r−)−ψT r+−ψT r−
cs r+,r−,B + − thedemandsidedoesn’tchangetheoriginallinearrelationship
s.t. d¯=¯b−Gπ, between the actual energy consumption and retail price; the
B =κ(B +τr+−r−/ρ), benefit of storage to the consumer side is in the form of
i+1 i i i
B =B ,0≤B ≤C, arbitrage options.
24 0 i
0≤r+ ≤ru,0≤r− ≤rd, Given the retail price π, let r(π) be the optimal charging
i i
vector by solving an equivalent of (17) and obtain Q(π).
where ψ and ψ are the costs associated charing and
+ − Withdecentralizedstorageintegration,theconsumersurplusis
discharging the battery, B the initial energy level in the
0 changedto cs (π)=cs(π)+Q(π), and the retailprofitwith
storage, C the capacity of the battery, ru the charging limit, consumer-basedds storageisrp (π)=rp(π)+πTr(π)−λ¯Tr(π).
and rd the discharging limit. ds
Byvaryingη ∈[0,1],maximizingtheretailpayofffunction
Notice that boththe objectivesand constraintscan be sepa-
will give us the tradeoff curve between CS and RP with
ratedforthepartwithpriceπandthepartwithstoragecontrol consumer-baseddecentralizedstorage.The followingtheorem
policyr, theretailprofitwithcentralizedstorage,rpcs(π), can characterizes the shape of the Pareto front of the CS-RP
be expressed as sum of the original retail profit, rp(π), and
tradeoffs in the presence of decentralized storage integration.
thearbitrageprofitQ(λ¯),i.e., rpcs(π)=rp(π)+Q(λ¯),where Theorem 4: Let the social welfare and retail profit maxi-
Q is defined as, mization prices with decentralized storage integration be πo
ds
Q(λ¯)=∆max −λ¯T(r+−r−)−ψTr+−ψTr− and πdsws, respectively.
r+,rs−.t,.B Bi+1 =κ(Bi+τri++−ri−/ρ−), (17) 1) Trehseulstioncgiarlewtaeillfparroefimt aisxirmpdisz(aλ¯ti)on=p0r.icTehiescπodsrwsre=spoλ¯n.dTinhge
B24 =B0,0≤Bi ≤C, consumer surplus is greater than that in absence of
0≤r+ ≤ru,0≤r− ≤rd. storage, i.e., cs (πsw)≥cs(πsw).
i i ds ds
2) AssumethatQ(π˜)>3Q(λ¯),whereπ˜ =G−1b,whichis
Since zero vector is a feasible solution, we have Q(λ¯)≥0.
the cut-off price resulting in zero demand. There exists
With the preferenceweight parameter on consumer surplus
a threshold ξ such that, when B ≤ ξ, the maximized
η, the retailer’s payoff function is
profit is less than that in absence of storage integration,
rpcs(π)+ηcs(π)=rp(π)+ηcs(π)+Q(λ¯). (18) i.e., rpds(πdos))<rp(πo) .
3) TheParetofrontwithdecentralizedintegrationisalways
Therefore,foranyη,theoptimalpricefortheretaileristhe inside that of the centralized integration with the same
same as in the case without storage, and the Pareto front is storage parameters.
shifted up by Q(λ¯).
Proof: When the preference weight factor on consumer
surplus η =1, the retailer’s payoff function is social welfare.
C. Analysis of Decentralized Storage Integration Notice that
sw (λ¯) =rp (λ¯)+cs (λ¯)=rp(λ¯)+cs(λ¯)+Q(λ¯)
We now consider decentralized storage integration by con- ds ds ds
≥rp(π)+cs(π)−λ¯Tr(π)−ψTr+(π)−ψTr−(π)
sumers. With storage and known DAHP, a consumer can + −
=rp(π)+cs(π)+Q(π)+πTr(π)−λ¯Tr(π),
optimize charging and discharging decisions. We assume that
consumers have the net-metering option that allows them to for any π, which means that the social welfare maximization
sell back surplus energy stored in the battery. But we restrict price is λ¯, resulted retail profitis 0, and the maximizedsocial
to the case that the storage capacity is below the level of welfare is increased by Q(λ). Therefore, the social welfare
aggregated demand, so that the retail price will always be maximization point is outside the original trade-off curve.
positive. We believe that this assumption is reasonable at When the preference weight factor η = 0, the retailer’s
current storage integration level. payofffunctionistheretailprofit.Withoutstorage,theoptimal
9
price is πo = (λ¯ +π˜)/2. With the decentralized consumer- 6
based storage, the consumer surplus is increased by Q(πo),
whiletheretailprofitischangedby−(πdos)Tr(πdos)−λ¯Tr(πddoss). 3)0 5
Also,wedenoteπo =πo+∆π.WeknowQ(π)isapiecewise (1 4
linear function ofdπs. Therefore, fit
o 3
r
Q(πo) = Q(π˜+λ¯/2+∆π), p no storage
ds ≥ Q(π˜)/2−maxπ{−λ¯Tr(π)}/2 etail 2 cuotinlitsyu−mbaesr−ebda ssteodra sgteorage
−max {−(∆π)Tr(π)}, (19) R 1
π
≥ Q(λ¯)+(Q(π˜)−3Q(λ¯))/2 −09 −8 −7 −6 −5 −4 −3 −2 −1
−maxπ{−(∆π)Tr(π)}. Consumer surplus (103)
Since Q(π˜)>3Q(λ¯)), there exists a threshold ξ s.t. when Fig. 8: The Pareto front with storage devices
the storage size is less than ξ, ∆π is small enough to make
Q(πo)>Q(λ¯). Therefore, the RP changes,
ds
(πo)Tr(πo)−λ¯Tr(πo) zero, and the consumer surplus increases by Q(λ¯). When
ds ds ds
≤−Q(πo)−ψTr+(πo)−ψTr−(πo)−λ¯Tr(πo) η = 0, the retailer maximizes its own profit. The leftmost
ds + ds − ds ds
<−Q(λ¯)−ψTr+(πo)−ψTr−(πo)−λ¯Tr(πo)≤0. pointonthenewParetofrontshowsthattheconsumersurplus
+ ds − ds ds increaseswhiletheretailprofitdecreasesata muchfasterrate
We have the maximized profit decreased with small storage. comparing with the original tradeoff curve. In this case, the
For a particular η ∈ (0,1), assume the optimal price with profit maximization point is inside the original Pareto front.
decentralized consumer-based storage is π (η). We use this
ds The Pareto front with consumer-based storage crosses with
price for the case with centralized utility-based storage. The
theoriginalParetofront.Thismeansthatonlywhentheretailer
change of social welfare is non-negative. The corresponding
is operating on the right of the cross point, consumer-based
point x has decreased consumer surplus and increased retail
storage can benefit the retail market. On the contrary, the
profit. Since the slope for that point is −η, all the left points
Pareto front with utility-based storage is outside the original
on the tradeoff curve with decentralized storage have less
Pareto front. Therefore, utility-based storage always benefits
social welfare, this point x is outside the trade-off curve with
the retail market.
consumer-basedstorage.Ontheotherhand,π (η)isafeasible
ds Another conclusion from Theorem 4 is that utility-based
priceforthecasewithcentralizedutility-basedstorageandthe
storage results in an enlarged Pareto front comparing with
corresponding point is inside the trade-off curve with utility-
the trade-off curve with consumer-based storage of the same
based storage. Therefore, the centralized utility-based storage
size. This conclusion can be also identified from Fig. 8.
results in a trade-off curve completely outside the one with
Comparing the trade-off curves with utility-based storage and
decentralized consumer-based storage. (cid:4)
consumer-based storage, we can see that the former one is
Theorem 4 shows that, under some conditions, maximized
completelyoutside thelatter, whichmeansthathavingutility-
retail profit with decentralized storage decreases while the
based storage brings more benefit to the retail market than
maximized social welfare increases, comparing with the case
consumer-based one, if no other considerations are taken.
without storage. Therefore, the new trade-off curve is inside
Furthermore, for utility-based storage, the optimal DAHP
theoriginalcurveontheleftsideandoutsidetheoriginalcurve
prices won’t change. While for consumer-based storage, we
ontherightside;thetwocurveswillintersectwitheachother,
plotted the profit maximization prices with different storage
similar to decentralized renewable integration.
implementationlevelsasshowninFig.9.Duetothearbitrage
Since demand level varies across different hours, the cut-
opportunity with consumer-based storage, the price becomes
off price π˜ will lead to high arbitrage opportunity.Therefore,
flatter. This shows that only consumer-based storage will
the condition Q(π˜) > 3Q(λ¯) is reasonable in practice. For
change the consumers’ energy consumption pattern.
the setup in our following simulation, the arbitrage profit
with cutoff price π˜ is much higher than with wholesale price,
Q(π˜)>Q(λ¯), i.e., the condition of Theorem 4 is satisfied.
0.6
Numerical simulation results are plotted in Fig. 8. For the
)
h
decentralizedcase,weassume20%oftheconsumershavethe W0.5
storage devices. For the centralizedmodel, we assume utility- K
$/
based storage has the same size as the aggregated consumer- (0.4
e
based storage to make fair comparison. c
Pri0.3
storage %50 implementation
ail storage %20 implementation
et0.2 no storage
Fig.8showsthatwhentheretailerhasaccesstothestorage R
devices, the Pareto front is shifted upward by Q(λ¯) and the 0.11 5 10 15 20 24
social welfare point becomes economically viable. Hour
Ontheotherhand,asfortheconsumer-basedstorage,when
theweightonconsumersurplusη =1,theretailprofitremains Fig. 9: Optimal DAHP with storage devices
10
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more complex situation to investigate separately. partment of Automation, Tsinghua University
in 2009. He is currently working toward the
Ph.D. degree in the School of Electrical and
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